GATE EC question library

Given $A=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$, the state transition matrix $e^{A t}$ is given by

Consider the signal $x(t)$ shown in the figure. Let $h(t)$ denote the impulse response of the filter matched to $x(t)$, with $h(t)$ being non-zero only in the interval 0 to $4 \; \mathrm{sec}$. The slope of $h(t)$ in the interval $3 < t < \text{sec}.$ is

A $1 \mathrm{~mW}$ video signal having a bandwidth of $100 \; \mathrm{MHz}$ is transmitted to a receiver through a cable that has $40 \mathrm{~dB}$ loss. If the effective one-sided noise spectral density at the receiver is $10^{-20}$ Watt $/ \mathrm{Hz}$, then the signal-to-noise ratio at the receiver is

A $100 \; \mathrm{MHz}$ carrier of $1 \mathrm{~V}$ amplitude and a $1 \; \mathrm{MHz}$ modulating signal of $1 \mathrm{~V}$ amplitude are fed to a balanced modulator. The output of the modulator is passed through an ideal high-pass filter with cut-off frequency of $100 \; \mathrm{MHz}$. The output of the filter is added with $100 \; \mathrm{MHz}$ …

Two sinusoidal signals of same amplitude and frequencies $10 \; \mathrm{kHz}$ and $10.1 \; \mathrm{kHz}$ are added together. The combined signal is given to an ideal frequency detector. The output of the detector is

Consider a binary digital communication system with equally likely $0$'s and $1$'s. When binary 0 is transmitted the voltage at the detector input can lie between the levels $-0.25 \mathrm{~V}$ and $+0.25 \mathrm{~V}$ with equal probability: when binary $1$ is transmitted, the voltage at the detector can have any value between $0$ and …

A random variable $\mathrm{X}$ with uniform density in the interval $0$ to $1$ is quantized as follows: $\begin{array}{ll} \qquad \text{If 0 < X < 0.3,} & \text{x}_{4} = 0 \\ \qquad \text{If 0.3 < X < 1,} & \text{x}_{4} = 0.7 \end{array}$ where $\text{x}_{4}$ is the quantized value of $\text{X}$. The root-mean square value of …

Choose the correct one from among the alternatives $\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}$ after matching an item from Group $1$ with the most appropriate item in Group $2.$ …

Three analog signals, having bandwidths $1200 \mathrm{~Hz}$, $600 \mathrm{~Hz}$ and $600 \mathrm{~Hz}$, are sampled at their respective Nyquist rates, encoded with $12$ bit words, and time division multiplexed. The bit rate for the multiplexed signal is

Consider a system shown in the figure. Let $X(f)$ and $Y(f)$ denote the Fourier transforms of $X(t)$ and $y(t)$ respectively. The ideal HPF has the cutoff frequency $10 \mathrm{kHz}$. The positive frequencies where $Y(f)$ has spectral peaks are

A parallel plate air-filled capacitor has plate area of $10^{-1} \mathrm{~m}^{2}$ and plate separation of $10^{-3} \mathrm{~m}$. It is connected to a $0.5 \mathrm{~V}, 3.6 \; \mathrm{GHz}$ source. The magnitude of the displacement current is $\left(\varepsilon_{0}=1 / 36 \pi \times 10^{-9} \mathrm{~F} / \mathrm{m}\right)$

A source produces binary data at the rate of $\text{10 kbps.}$ The binary symbols are represented as shown in the figure given below. The source output is transmitted using two modulation schemes, namely Bìnary PSK (BPSK) and Quadrature PSK (QPSK). Let $\mathrm{B}_{1}$ and $\mathrm{B}_{2}$ be the bandwidth requirements of BPSK and QPSK respectively. …

Consider a $300 \; \Omega$, quarter-wave long (at $1 \; \mathrm{GHz}$ ) transmission line as shown in the figure. It is connected to a $10 \mathrm{~V}, 50 \; \Omega$ source at one end and is left open circuited at the other end. The magnitude of the voltage at the open circuit end of the line is

In a microwave test bench, why is the microwave signal amplitude modulated at $1 \; \mathrm{kHz}$?

If $\overrightarrow{\mathrm{E}}=\left(\hat{a}_{x}+j \vec{a}_{y}\right) e^{\beta a-k a t}$ and $\overrightarrow{\mathrm{H}}=\left(\frac{k}{\omega \mu}\right)\left(\hat{a}_{y}+j \hat{a}_{x}\right) e^{j u-\mu \mu 1}$, the time averaged Poynting vector is

Consider an impedance $\mathrm{Z = R + j X }$ marked with point $\mathrm{P}$ in an impedance Smith chart as shown in the figure. The movement from point $\mathrm{P}$ along a constant resistance circle in the clockwise direction by an angle $45^{\circ}$ is equivalent to

A plane electromagnetic wave propagating in free space in incident normally on a large slab of loss-less, non magnetic, dielectric material with $\varepsilon>\varepsilon_{0}$. Maxima and minima are observed when the electric field is measured in front of the slab. The maximum electric field is found to be $5$ times the minimum field. The intrinsic …

A lossless transmission line is terminated in a load which reflects a part of the incident power. The measured VSWR is $2.$ The percentage of the power that is reflected back is

A signal is sampled at $8 \; \mathrm{kHz}$ and is quantized using $8$-bit uniform quantizer. Assuming $\mathrm{SNR}_q$ for a sinusoidal signal, the correct statement for $P C M$ signal with a bit rate of $R$ is

The minimum number of equations required to analyze the circuit shown in the figure is